On almost quotient Yamabe solitons (Q6648694)

The authors investigate almost quotient Yamabe solitons which are special solutions of the nonlinear flow \N\[\N\frac{\partial{g}}{\partial t}(t)=-\left( \log\frac{\sigma_{k}(g(t))} {\sigma_{l}(g(t))} -\frac{\int_{M}\sigma_{l}(g(t)) \log\frac{\sigma_{k}(g(t))} {\sigma_{l}(g(t))} dv_{g(t)} }{\int_{M}\sigma_{l}(g(t))dv_{g(t)} }\right)g(t),\,\,\,g(0)=g_{0},\N\]\Nwhere \(\sigma_{k}\) is the kth elementary symmetric function of the eigenvalues of the endomorphism \(\frac{1}{n-2}g^{-1}(\mathrm{Ric}-\frac{R}{2(n-1)}g)\). The authors give some examples of almost gradient quotient Yamabe solitons and almost gradient quotient Yamabe solitons. They provide some conditions for a complete almost quotient Yamabe soliton to become trivial or isometric to an Euclidean sphere. For instance, they prove that any compact quotient Yamabe soliton with a vanishing Cotton tensor is trivial. Also, they establish that any compact quotient gradient Yamabe soliton is trivial.